Proposition 2.5 ([1]} Proposition 3.15) 
Let \((\mathcal {B}_1,|\cdot |_1)\) , \((\mathcal {B}_2,|\cdot |_2)\)  be Banach spaces.
For any \(k\ge 1\)  and any \(M>1\)  there exist \(L,\delta \)  such that,
for every two injective bounded linear operators \(A_1,A_2:\mathcal {B}_1 \rightarrow \mathcal {B}_2\)  and two \(k\) -dimensional subspaces \(E_1,E_2\)  of \(\mathcal {B}_1\)  satisfied that
\(|A_j|&, |(A_j|_{E_j})^{-1}|\le M \quad \text{for} \ j=1,2, \\|A_1-A_2|&, d_H(E_1,E_2)\le \delta ,\) 
